Page 495 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

τριπλασίονι λόγῳ εἰσὶ τῶν ἐν ταῖς βάσεσι διαμέτρων· ὅπερ thing was shown (to be) impossible. Thus, cone ABCDL
ἔδει δεῖξαι. does not have to some solid greater than cone EFGHN
the cubed ratio than BD (has) to FH. And it was shown
that neither (does it have such a ratio) to a lesser (solid).
Thus, cone ABCDL has to cone EFGHN the cubed ra-
tio that BD (has) to FG.
igþ the same base as the cone, and of the same height as it
And as the cone (is) to the cone, so the cylinder (is)
to the cylinder. For a cylinder is three times a cone on

[Prop. 12.10]. Thus, the cylinder also has to the cylinder
the cubed ratio that BD (has) to FH.
Thus, similar cones and cylinders are in the cubed ra-
tio of the diameters of their bases. (Which is) the very
thing it was required to show.

.
Proposition 13
R A H G T F
᾿Εὰν κύλινδρος ἐπιπέδῳ τμηθῇ παραλλήλῳ ὄντι τοῖς ἀπε-
If a cylinder is cut by a plane which is parallel to the
ναντίον ἐπιπέδοις, ἔσται ὡς ὁ κύλινδρος πρὸς τὸν κύλινδρον, opposite planes (of the cylinder) then as the cylinder (is)
E Z X P R A G C T M
οὕτως ὁ ἄξων πρὸς τὸν ἄξονα. to the cylinder, so the axis will be to the axis.
V
E
N
L
F
O
K
S B D U
For let the cylinder AD have been cut by the plane
Κύλινδρος γὰρ ὁ ΑΔ ἐπιπέδῳ τῷ ΗΘ τετμήσθω πα- Q S B H D U W
ραλλήλῳ ὄντι τοῖς ἀπεναντίον ἐπιπέδοις τοῖς ΑΒ, ΓΔ, καὶ GH which is parallel to the opposite planes (of the cylin-
συμβαλλέτω τῷ ἄξονι τὸ ΗΘ ἐπίπεδον κατὰ τὸ Κ σημεῖον· der), AB and CD. And let the plane GH have met the
λέγω, ὅτι ἐστὶν ὡς ὁ ΒΗ κύλινδρος πρὸς τὸν ΗΔ κύλινδρον, axis at point K. I say that as cylinder BG is to cylinder
οὕτως ὁ ΕΚ ἄξων πρὸς τὸν ΚΖ ἄξονα. GD, so axis EK (is) to axis KF.
᾿Εκβεβλήσθω γὰρ ὁ ΕΖ ἄξων ἐφ᾿ ἑκάτερα τὰ μέρη ἐπὶ For let axis EF have been produced in each direction
τὰ Λ, Μ σημεῖα, καὶ ἐκκείσθωσαν τῷ ΕΚ ἄξονι ἴσοι ὁσοι- to points L and M. And let any number whatsoever (of
δηποτοῦν οἱ ΕΝ, ΝΛ, τῷ δὲ ΖΚ ἴσοι ὁσοιδηποτοῦν οἱ ΖΞ, lengths), EN and NL, equal to axis EK, be set out (on
ΞΜ, καὶ νοείσθω ὁ ἐπὶ τοῦ ΛΜ ἄξονος κύλινδρος ὁ ΟΧ, the axis EL), and any number whatsoever (of lengths),
οὗ βάσεις οἱ ΟΠ, ΦΧ κύκλοι. καὶ ἐκβεβλήσθω διὰ τῶν FO and OM, equal to (axis) FK, (on the axis KM).
Ν, Ξ σημείων ἐπίπεδα παράλληλα τοῖς ΑΒ, ΓΔ καὶ ταῖς And let the cylinder PW, whose bases (are) the circles
βάσεσι τοῦ ΟΧ κυλίνδρου καὶ ποιείτωσαν τοὺς ΡΣ, ΤΥ PQ and V W, have been conceived on axis LM. And
κύκλους περὶ τὰ Ν, Ξ κέντρα. καὶ ἐπεὶ οἱ ΛΝ, ΝΕ, ΕΚ let planes parallel to AB, CD, and the bases of cylinder
ἄξονες ἴσοι εἰσὶν ἀλλήλοις, οἱ ἄρα ΠΡ, ΡΒ, ΒΗ κύλινδροι PW, have been produced through points N and O, and
πρὸς ἀλλήλους εἰσὶν ὡς αἱ βάσεις. ἴσαι δέ εἰσιν αἱ βάσεις· let them have made the circles RS and T U around the
ἴσοι ἄρα καὶ οἱ ΠΡ, ΡΒ, ΒΗ κύλινδροι ἀλλήλοις. επεὶ οὖν centers N and O (respectively). And since axes LN, NE,
οἱ ΛΝ, ΝΕ, ΕΚ ἄξονες ἴσοι εἰσὶν ἀλλήλοις, εἰσὶ δὲ καὶ οἱ and EK are equal to one another, the cylinders QR, RB,
ΠΡ, ΡΒ, ΒΗ κύλινδροι ἴσοι ἀλλήλοις, καί ἐστιν ἴσον τὸ and BG are to one another as their bases [Prop. 12.11].
πλῆθος τῷ πλήθει, ὁσαπλασίων ἄρα ὁ ΚΛ ἄξων τοῦ ΕΚ But the bases are equal. Thus, the cylinders QR, RB,
ἄξονος, τοσαυταπλασίων ἔσται καὶ ὁ ΠΗ κύλινδρος τοῦ ΗΒ and BG (are) also equal to one another. Therefore, since
κυλίνδρου. διὰ τὰ αὐτὰ δὴ καὶ ὁσαπλασίων ἐστὶν ὁ ΜΚ ἄξων the axes LN, NE, and EK are equal to one another,
τοῦ ΚΖ ἄξονος, τοσαυταπλασίων ἐστὶ καὶ ὁ ΧΗ κύλινδρος and the cylinders QR, RB, and BG are also equal to one
τοῦ ΗΔ κυλίνδρου. καὶ εἰ μὲν ἴσος ἐστὶν ὁ ΚΛ ἄξων τῷ another, and the number (of the former) is equal to the
ΚΜ ἄξονι, ἴσος ἔσται καὶ ὁ ΠΗ κύλινδρος τῷ ΗΧ κυλίνδρῳ, number (of the latter), thus as many multiples as axis KL

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